2.8 Appendix: Bounds for Special Functions
3.1.1 Statement of Result
A basic problem in modern number theory and the analytic theory of modular forms is to understand the limiting behavior of modular forms in families. Let f : H→ C be a classical holomorphic newform of weight k and level q. The mass of f is the finite measure dνf =
|f(z)|2yk−2dx dy(z=x+iy) on the modular curveY0(q) = Γ0(q)\H. In a recent breakthrough, Holowinsky and Soundararajan [25] proved that newforms of large weight k and fixed level q = 1 have equidistributed mass, answering affirmatively a natural variant1 of the quantum unique ergodicity conjecture of Rudnick and Sarnak [52].
Theorem 3.1.1 (Mass equidistribution for SL(2,Z) in the weight aspect). Let f traverse a sequence of newforms of increasing weight k → ∞ and fixed level q = 1. Then the mass νf
equidistributes2 with respect to the Poincar´e measure dµ = y−2dx dy on the modular curve Y0(q).
1as spelled out by Luo and Sarnak [42]; we refer to Sarnak [53, 54] and the references in [25]
for further discussion.
2We say that a sequence of finite Radon measuresµjon a locally compact Hausdorff spaceX equidistributeswith respect to some fixed finite Radon measureµif for each functionφ∈Cc(X) we haveµj(φ)/µj(1)→µ(φ)/µ(1) asj→ ∞, here and always identifying a measure µwith the corresponding linear functionalφ7→µ(φ) :=R
Xφ dµ on the spaceCc(X) and writing 1 for the constant function.
Kowalski, Michel, and VanderKam [36, Conj 1.5] formulated an analogue of the Rudnick- Sarnak conjecture in which the roles of the parametersk andq are reversed: they conjectured that the masses of newforms of fixed weight and large level qare equidistributed amongst the fibers of the canonical projectionπq :Y0(q)→Y0(1) in the following sense.
Conjecture 3.1.2 (Mass equidistribution for SL(2,Z) in the level aspect). Let f traverse a sequence of newforms of fixed weight and increasing levelq→ ∞. Then the pushforwardµf :=
πq∗(νf)of the mass of f toY0(1) equidistributes with respect toµ.
Kowalski, Michel and VanderKam remark that Conjecture 3.1.2 follows in the special case of dihedral forms from their subconvex bounds for Rankin-SelbergL-functions modulo an unestab- lished extension of Watson’s formula [70], which is now known by theorem 3.4.1 of this chapter.
Recently Koyama [37], following the method of Luo and Sarnak [41], proved the analogue of Conjecture 3.1.2 for unitary Eisenstein series of increasing prime level by reducing the problem to known subconvex bounds for automorphicL-functions of degree two.
Our aim in this chapter is to establish the squarefree level case of Conjecture 3.1.2. Our result is the first of its kind for nondihedral cusp forms.
Theorem 3.1.3 (Mass equidistribution for SL(2,Z) in the squarefree level aspect). Let f tra- verse a sequence of newforms of fixed weight and increasing squarefree level q → ∞. Thenµf equidistributes with respect toµ.
Remark 7. Our extension (theorem 3.4.1) of Watson’s formula [70] shows that theorem 3.1.3 would follow from subconvex bounds L(f ×f ×φ,1/2) φ q1−δ (δ > 0) for the central L- values of the triple productL-functions attached to f as above and each Maass cusp form or unitary Eisenstein series φ on Y0(1). Such bounds are known to follow from the generalized Lindel¨of hypothesis, which itself follows from the generalized Riemann hypothesis, so one can view theorem 3.1.3 as an unconditionally proven consequence of a central unresolved conjecture.
Remark 8. One cannot relax entirely the restriction of theorem 3.1.3 to newforms, since for instance a cusp form of level 1 may be regarded as an oldform of arbitrary levelq >1.
Remark 9. Rudnick [51] showed that theorem 3.1.1 implies that the zeros of newforms of level 1 and weightk→ ∞equidistribute onY0(1). At the 2010 Arizona Winter School, Soundararajan asked whether there is an analogue of Rudnick’s result for newforms of large level. We do not know whether such an analogue exists and highlight here one of the difficulties in adapting Rudnick’s method. Letf be a newform of weight kand levelq, letZ be the left Γ0(q)-multiset of zeros of f in Hand letZ1 be the left Γ-multiset (Γ = PSL(2,Z)) obtained by summing the images ofZ under coset representatives for Γ(1)/Γ0(q). We ask: does Γ\Z1 equidistribute on
Y0(1) as q → ∞? Following Rudnick, one may show for φ∈Cc∞(H) and Φ(z) =P
γ∈Γφ(γz) that
12 kψ(q)
X
z∈Γ\Z1
Φ(z)
# StabΓ(z)= Z
Γ\H
ΦdV + Z
Γ\H
πq∗(logνf)
kψ(q) ∆ΦdV, (3.1)
whereψ(q) = [Γ(1) : Γ0(q)], ∆ =y2(∂x2+∂y2) is the hyperbolic Laplacian, anddV is the hyper- bolic probability measure on Γ\H; the formula (3.1) follows by some elementary manipulations of the identityR
Hlog|z−z0|∆φ(z)y−2dx dy= 2πφ(z0), which holds for anyz0∈Hand follows from Green’s identities. Since the total number of inequivalent zeros is #Γ\Z1= #Γ0(q)\Z ∼ kψ(q)/12 [60,§2], the first term on the right-hand side of (3.1) may be regarded as a main term, the second as an error term that one would like to show tends to 0. An important step toward adapting Rudnick’s method would be to rule out the possibility thatπq∗(logνf)/kψ(q) tends to
−∞uniformly on compact subsets as q→ ∞. The difficulty in doing so is that theorem 3.1.3 does not seem to preclude the massesνf from being very small somewhere within each fiber of the projectionY0(q)→Y0(1); stated another way, the sum of the values taken byyk|f|2 in a fiber ofY0(q)→Y0(1) are controlled (in an average sense as the fiber varies) by theorem 3.1.3, but theirproductcould still conceivably be quite small. There are further difficulties in adapting Rudnick’s method that we shall not mention here.
Remark 10. Lindenstrauss [40] and Soundararajan [65] proved that Maass eigencuspforms of fixed levelqand large Laplace eigenvalueλ→ ∞have equidistributed mass. We ask: do Maass newforms of large level q → ∞ (with λ taken to lie in a fixed subinterval of [1/4,+∞], say) satisfy the natural analogue of Conjecture 3.1.2? An affirmative answer to this question would follow from the generalized Riemann hypothesis (at least forq squarefree, as in remark 7), but appears beyond the reach of our methods because the Ramanujan conjecture is not known for Maass forms (compare with [25, p.2]).
Remark 11. We shall actually establish the following stronger hybrid equidistribution result: for a newformf of (possiblyvarying) weightk and squarefree levelq, the measures µf =πq∗(νf) equidistribute asqk→ ∞. The novelty in our argument concerns only the variation ofq, so we encourage the reader to regardkas fixed.
Remark 12. With minor modifications our arguments should extend to the general case of not necessarily squarefree levelsqas soon as an appropriate extension of Watson’s formula is worked out. However, we shall invoke the assumption that the levelq is squarefree whenever doing so simplifies the exposition. The parts of our argument that require modification to treat the general case are Lemmas 3.3.4, 3.3.13, and 3.4.3. One should be able to generalize Lemmas 3.3.4 and 3.3.13 using that for any level q the cusps of Γ0(q) fall into classes indexed by the divisorsdofqconsisting ofφ(gcd(d, q/d)) cusps of widthd/gcd(d, q/d). To generalize 3.4.3, one
must compute (or sharply bound) ap-adic integral involving matrix coefficients of supercuspidal representations of GL(2,Qp). We plan to consider this generalization in future work.